Sarkaria's non-embeddability conjecture for 2-complexes in four-space

Let $K$ be a $2$-dimensional simplicial complex, and let $f_i(K)$ denote the number of its $i$-simplices.

Sarkaria's non-embeddability conjecture. If

$$f_2(K) \geq 4 f_1(K),$$

then $K$ cannot be embedded into $\mathbb{R}^{4}$.

This conjecture concerns the extremal number of triangles in a complex embeddable in four-dimensional Euclidean space. The source attributes it to Sarkaria through Kalai's blog, but gives no resolution.